User:Userminusone/Averiant

The process of taking the averiant between two ratios is a variant of taking the mediant between two ratios which was invented by me (Userminusone). The averiant between two ratios is reasonably close to the geometric mean between the same two ratios, which is the same as the average between the cent values of the two ratios, hence the name. (average + mediant = averiant)

Calculation

To calculate the averiant between a/b and c/d, first multiply a and b by (c+d) to get a(c+d)/b(c+d), then multiply c and d by (a+b) to get c(a+b)/d(a+b), then take the mediant between the resulting two fractions to get (a(c+d)+c(a+b))/(b(c+d)+d(a+b)). This can be simplified to (ad + bc + 2ac)/(ad + bc + 2bd).

(x/y)Averiants, Averiantal percentages, and VDn

The averiant can be generalized by adding an (x/y) parameter which denotes how far from a/b to c/d the resulting averiant is. For example, the (2/3)averiant between 4/3 and 2/1 would be approximately 2/3 of the way from 4/3 to 2/1, in terms of its cent value. To calculate the (x/y)averiant between a/b and c/d, multiply a and b by (y-x)(c+d) to get (a(y-x)(c+d))/(b(y-x)(c+d)), multiply c and d by x(a+b) to get (cx(a+b))/(dx(a+b)), and then take the mediant between the resulting two fractions, which simplifies to (ay(c+d)+x(bc-ad))/(by(c+d)-x(bc-ad)).

The process of finding the (x/y)averiant between a/b and c/d has an inverse: finding the (a/b - c/d)averiantal percentage of e/f. (This finds the percentage, x/y, given that the (x/y)averiant between a/b and c/d equals e/f) The formula for the (a/b - c/d)averiantal percentage of e/f is as follows: x/y = ((be-af)(c+d))/((e+f)(bc-ad))

While EDn stands for Equal Divisions of the interval n, VDn stands for aVeriantal Divisions of the interval n. (V is used in place of A because ADO is already defined as Arithmetic Divisions of the Octave)

mVDn is generated by taking all the (x/y)averiants between 1/1 and n where y = m. (in this case, x ranges from 1 to m, and the resulting percentages used to generate the averiants are 1/m, 2/m, 3/m ... (m-1)/m, m/m)

Examples

The (1/2)averiant between 1/1 and 2/1 is 7/5

The (1/3)averiant between 1/1 and 2/1 is 5/4

The (1/5)averiant between 1/1 and 2/1 is 8/7

The (1/1 - 2/1)averiantal percentage of 3/2 is 3/5

The (1/1 - 2/1)averiantal percentage of 4/3 is 3/7

The (1/1 - 2/1)averiantal percentage of 16/15 is 3/31

8VDO represents the scale: (25/23, 13/11, 9/7, 7/5, 29/19, 5/3, 31/17, 2/1)

9VDF aka 9VD3/2 (which could be thought of as a JI version of Carlos Alpha) represents the scale: (23/22, 47/43, 8/7, 49/41, 5/4, 17/13, 26/19, 53/37, 3/2)

(note that 1200log2(e/f) is always relatively close to, but flatter than 1200*x/y. For example, 1200log2(8/7) is ~231.174 and 1200*1/5 = 240)